The reasons for the establishment of the League of Nations can bee seen in the year in which it was established: just after the first World War.
At that time, a big proportion of world was devastated by the effects of the war, and arguably almost the whole world was affected in some way. People seeked peace and wanted to establish means that would prevent a similar war in the future. An international organisation of nations was established with that purpose: to support world peace.
As we know, this aim was not reached, as soon another world-scale war followed. In fact by that time (1939), a lot of nations, including Germany (which in any case joined late, as it was not allowed to join just after they had lost another war) have withdrawn from the League of nations. Additionally, the League was not as powerfull as it would have been if the US had joined: another factor that contributed to its failure.
